On the fix-points of composite transcendental entire functions

Abstract

A transcendental entire function F(z) is called composite iff F(z) =f( g(z)) for some nonlinear entire functionsf and g. A point z0 in the complex plane is called a fix-point of F iff F(z,) = zo. Gross [3] conjectured that every composite transcendental entire function must have infinitely many fix-points. It has been shown, thus far, that the conjecture is valid for finite order entire functions by several authors (e.g., [Z, 4, 71) via different approaches. In a previous paper [ 111, we investigated the fixpoints of a certain type of infinite order composite entire function. There, it is required that g be an entire function of positive finite order with a finite Nevanlinna exceptional value and ,f be an arbitrary entire function such that the hyperorder off(g) is less than the order of g. In this note we shall treat some other class of composite entire functions. The tools are, as usual, based on Nevanlinna’s theory of meromorphic functions and an elegant result due to Steinmetz [S, Corollary I]. Our result is the following theorem.